Sách Proof of the combinatorial nullstellensatz over integral domains, in the spirit of Kouba

Abstract
It is shown that by eliminating duality theory of vector spaces from a recent proof
of Kouba [A duality based proof of the Combinatorial Nullstellensatz, Electron. J.
Combin. 16 (2009), #N9] one obtains a direct proof of the nonvanishing-version
of Alon’s Combinatorial Nullstellensatz for polynomials over an arbitrary integral
domain. The proof relies on Cramer’s rule and Vandermonde’s determinant to
explicitly describe a map used by Kouba in terms of cofactors of a certain matrix.
That the Combinatorial Nullstellensatz is true over integral domains is a well-
known fact which is already contained in Alon’s work and emphasized in recent
articles of Micha lek and Schauz; the sole purpose of the present note is to point out
that not only is it not necessary to invoke duality of vector spaces, but by not doing
so one easily obtains a more general result.